Modeling of Corneal Biomechanical Properties and Responses Caveats for the prudent and thoughtful his section might better be entitled: “The Dutch Uncle Speaks” (and about time too) because there is an unfortunate tendency amongst our ophthalmic brethren to credit “science” over art and to ignore clinical experience. That is to say that x + y must always equal z because science says so — anything else is looked down on (and with a sneer) as anecdotal. [1] As a foot–slogger of long experience I have been sufficiently pummeled by the realities of medicine to tell you flatly that “it ain’t necessarily so”. The natterings of rear echelon commandos do not signify. When I first began to teach radial keratotomy, I felt that it was incumbent upon me to explain that the response to the surgery may not be what the surgeon expects. I had to remind erstwhile refractive surgeons of the old cliché: “Man proposes and God (nature) disposes”. In short — what happens to the eye happens and we must live with that — all that we’ve been taught to the contrary. Basically it meant that the budding refractive surgeon must be of Zen to be successful. It meant that one had to accept the fact that there is no “should” or “ought to be”; there is only what is. That is not to say that there is no cause and effect — that there surely is. It is merely to say that the cause portion of the equation may not be as you suppose. Experiments in modern Gestalt psychology demonstrate a universal need to discover familiar patterns in apparently random occurrences; to bring order out of chaos; to explain the unexplainable. But this quest for comfort and security sometimes leads to bizarre conclusions: The cock crows at daybreak and the sun also rises. Hence the sun rises in response to the crowing of the cock. Ridiculous to be sure but it was once earnestly believed, supported by sublime and elegant proofs which adequately explained most observed phenomena, that the sun rotated around the Earth. We have subsequently discovered another reality. In the early days of RK there were encounters that caused the author great consternation. Frequently I would find myself approached by some shiny–faced youth (occasionally not so shiny–faced) who would introduce him or herself and then immediately set to explaining how RK worked, and of course they had it all figured out because they held a degree in engineering. [2] Most of their theories were amusing but some were downright terrifying. Fortunately, truth has a way of burning away fancy if you let it. Still the tendency to credit vapid musings over nitty–gritty reality persists especially when garnished with numbers — preferably arranged into complex equations. We seem to have forgotten (if we indeed ever knew it) that mathematics is an artificial construct and that an inch — isn’t. Mathematical expressions may be useful in explaining some of the phenomena we experience in this singularity but they are not real. They have been bent, twisted and bludgeoned until they concur with real–life experiences — sometimes. If we are lucky this happens most of the time. But sometimes it doesn’t. Sometimes the product of mathematical manipulation is a phantasm. And all to often the prestidigitators who deal with such obscurities accept and preach these results as holy writ. We humans also have a tendency to bestow a certain cachet on those who seem in mastery of a discipline with which we have little experience. Hence the awe accorded to physicians, the clergy, and rich folks. It must be remembered, though, that all of these people put their drawers on — one leg at a time. Then too it is said that “there are none so self–righteous as the converted sinner”. This would certainly apply to those among us who have recently acquired “wealth” in the form of new technology (read toys). Such a thing causes a burning desire to proselytize — to spread the word — often before the true situation becomes completely understood and often ending in causing great harm. Case in point — topography of the cornea (see also Beware the Instant Corneal Topography “Expert”). Well then, if all this science is mucking up the picture why should we give our attention and time to this business of biomechanics? The answer is: because the wise man will consider all things and file away those that do not fit. How does one know which things to keep and which to file, master? Well, grasshopper, we can mimic the process paleontologists follow to reconstruct a dinosaur — a creature never seen by man. They simply gather together all the parts, discard those which do not look like a Thingamasaurus, and assemble the rest. It may be thought that I am merely being droll, but that is exactly what paleontologists do. Of course like all magicians — they cheat. They have prepared themselves to perform this feat of legerdemain in advance by studying the bones of modern creatures and the way the muscles and tendons attach to these bones and the effect on bone growth of different stimuli. Hence they have provided themselves with the tools necessary to sort out that which is important and that which is not; and they never throw anything away. Note that earlier I said “file” away not throw away. Can we do less? Certainly not — this is what this section is about: providing you with the tools to assemble your own Thingamasaurus out of all the bits you will encounter in this new field. The study of biomechanics is vital to carrying forward our surgical techniques. It is just that we must apply what we discover with great caution. More importantly we needs must seek out those relationships and properties which best describe the true situation. In constructing that description, the best language to use is that of mathematics. But we must be prepared to coin new terms and expressions to use when it is evident that the old ones will not do. Making Sense of Biomechanical Modeling While it is possible to model aspects of corneal behavior based upon our knowledge of certain corneal properties — some observations and cautions regarding these computer models are in order: do not generalize from the results of modeling Unfortunately, many authors do just that. They assume that since their modeling matches some clinical observations that — ergo sum — their modeling must be correct. This is a dangerous assumption to be made. It leads to the equally erroneous conclusion that since certain clinical observations are not supported by their model that the clinical observations are wrong. The author would like to point out the only “universal truth” of refractive surgery: Eye — 12, Theory — 0. That is to say that the eye has defied and confounded theorists for years. This has been evident from the outset, especially as regards radial keratotomy. Hence the second observation is: credit clinical observations before theoretical notions One of the assumptions made in modeling is that shear effects are minimal in the cornea. This is patently not the case as evidenced by the internal corneal folds described by Bores, Fyodorov, and Seiler as well as others.[51, 52] Certain computer models have made untenable assumptions based upon expediency. It takes a great deal of effort to set up a corneal model. Because exact data establishing certain corneal properties are not known — they are either ignored or assumed to be something they are not. For example it is often assumed that the cornea is isotropic — that is, like steel, it has equal strength in any direction. This assumption might make modeling easier but since it is not true any assumptions made from a model based upon that supposition must be suspect — to say the very least; the author prefers the term — bogus. Some of this notion comes from work done on sclera. Unfortunately, the properties of the sclera are not those of the cornea. The sclera, for example, is not bounded by a relatively inelastic and homogeneous layer on one side (Bowman’s layer) and an elastic and slightly unhomogeneous layer on the other (Descemet’s). For another, the cornea is transparent whereas the sclera is not. This has important implications not the least of which is that the corneal collagen fibers are packed closer together — less than ½ wavelength of light. Another assumption that is often made is that the stroma is composed of successive layers all of which are parallel to the corneal surface and which slide freely over one another like stacked paper. This is close to the truth and may actually be the case except that it does not accord with clinical and histological evidence. Furthermore, it implies that the stroma is subject to shear — a fact that some investigators ignore. Absent Bowman’s layer and separated from limbal support, the stroma does illustrate shear — clinically. It is naive to ignore the extremely careful observations of Kokott regarding the direction and interweaving of the stromal keratocytes. X–ray diffraction analysis has revealed the organization of the collagen fibrils in human stroma, which run in a supero-inferior and medio-lateral direction in the central and paracentral cornea, and in a circumferential direction in the peripheral cornea.[53] While watching some experiments with peeling forces, the author observed that as the tissue peeled away from the underlying stroma, there was some “stutter” or irregularity in the strain–gauge readings as the tissue came away. While not conclusive, this could be caused by cross–linked tissue being broken. At the very least it indicates that the stroma is definitely unhomogeneous. Histological evidence bears this out in that the more superficial keratocytes are packed somewhat more loosely than those in the deeper layers. On the other hand, the sclera, except in certain areas such as under the rectus muscle insertions, is of fairly uniform thickness. This the cornea is decidedly not — yet most models assume that the cornea has a uniform thickness. In general, determination of the smallest homogeneous component of a composite material will result in the most accurate determination of material property. Conversely, such modeling should always be compared to gross material properties to avoid errors which might creep into these more complex analyses. The importance of constitutive properties cannot be overstated since they provide a vital link to reality in the finite element modeling of a real-world situation. The finite element model can be seen as a large “black box” into which geometric and constitutive properties are placed and from which calculations are generated of force and position. Despite impressive geometric correlation with reality, lack of appropriate constitutive properties will doom a finite element model to mediocre performance and distance it from reality. When evaluating a finite element model, therefore, the reader should focus on several important points, the accuracy of input parameters including constitutive properties, geometry and the fit to reality — that is, how well the model describes known behavior. For example: Hanna et al found that the mechanical model predicted that more elastic corneas would give a greater effect — a prediction at odds with clinical experience.[54] Any calculations should be performed in the nonlinear mode, and viscoelastic properties should be considered. For many corneal operations, nonaxisymmetric three–dimensional analysis will be required and appropriate selection of elements should ideally reflect internal corneal structure including the low bending moment of the cornea. By comparing these models to clinical experiments, useful models can be created which can significantly improve our understanding and predictability of corneal procedures. Modeling To develop a mathematical model of the cornea, certain components must be identified and employed: linear, nonlinear, or viscoelastic mathematical laws that describe the behavior of the cornea mechanical constants and constitutive physical properties, such as Young’s modulus boundary conditions suitable computer programs to solve equations and generate data comprehensive data interpretation construction of a three-dimensional, geometric presentation of the cornea Three steps are involved in the creation of a finite element model. The first, generation of a mesh, involves approximating the geometry of the problem with finite elements. Yang and Hoeltzel have recently performed an excellent review of finite element mesh generation schemes which provides an introduction to this subject.[55] Asymmetry implies that the problem is the same as one sweeps around a given axis and implies that only a two–dimensional finite element mesh is required. Radial keratotomy is an example of a nonaxisymmetric problem which requires the generation of a full three–dimensional mesh. The second step in the formulation of a finite element model is the solution of the finite element equations. This requires the input of constitutive properties. Finally, post processing is performed to provide both graphic and numerical output to show the results of the calculations. In some computer programs, these three steps are combined while others require separate programs for each step. Excellent texts are available and one book even describes simple problems which can be implemented in the Basic programming language (see Suggested Reading List).[56] Table 3 lists several computer programs available for personal computers. While the computation time can be time–consuming, even with a fast system (recommended), calculations using the finite element programs mentioned above are within the capabilities of anyone with a computer. Table 3. Modeling software [1]   Anecdotal is defined in the Oxford English Dictionary as “things unpublished” or “hitherto private or unpublished narrative or details of history.” It is often used disparagingly by a certain group of individuals to discredit careful clinical observations by “outsiders” and is part and parcel of the NIH (Not Invented Here) Syndrome. [2]    I once witnessed an electrical “engineer” damn near electrocute himself because he had overlooked the fact that a sweaty sock inside a rubber–soled shoe with a hole in it provides an excellent electrical pathway to a wet floor. © Leo D. Bores, MD - 2002